Hamiltonian operator total spin

Thus, the total Hamiltonian operator H for a hydrogen atom including spin-orbit coupling is... 

While the Hamiltonian operator Hq for the hydrogen atom in the absence of the spin-orbit coupling term commutes with L and with S, the total Hamiltonian operator H in equation (7.33) does not commute with either L or S because of the presence of the scalar product L S. To illustrate this feature, we consider the commutators [L, L S] and [S, L S],... 

H is the total Hamiltonian (in the angular frequency units) and L is the total Liouvillian, divided into three parts describing the nuclear spin system (Lj), the lattice (Ll) and the coupling between the two subsystems (L/l). The symbol x is the density operator for the whole system, expressible as the direct product of the density operators for spin (p) and lattice (a), x = p <8> ci. The Liouvillian (Lj) for the spin system is the commutator with the nuclear Zeeman Hamiltonian (we thus treat the nuclear spin system as an ensemble of non-interacting spins in a magnetic field). Ll will be defined later and Ljl... 

S denotes a vector operator comprising three components Sx, Sy, and Sz. Note that generally a spin Hamiltonian replaces all the orbital coordinates required to define the system by spin coordinates. With the total spin operator S = SA + SB and S2 = SA + Sg + 2SaSb, the Hamiltonian can be rewritten as... 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

The Hamiltonian (1) is spin free, commutative with the spin operator S2 and its z-component Sz for one-electron and many-electron systems. The total spin operator of the hydrogen molecule relates to the constituent one-electron spin operators as... 

Coulomb interactions dominate the electronic structure of molecules. The total spin S2 and Sz are nearly conserved for light atoms. We will consider spin-independent interactions in models with one orbital per site. In the context of tt electrons, the operators a+a and OpCT create and annihilate, respectively, an electron with spin a in orbital p. The Hiickel Hamiltonian is... 

Starting from n)> we notice that it is neither an eigenstate of a Heisenberg Hamiltonian nor an eigenstate of the total spin operator S2. For instance, the Neel state is not invariant under the action of the nearest-neighbor XY spin terms of the Heisenberg Hamiltonian, which may produce a spin-flip on two nearest-neighbor sites at a time with respect to n). These XY excitation operators from the vacuum 4>n > can be written as... 

Since the Heisenberg Hamiltonian commutes with the total-spin operators, the 22iV-dimensional spin space can be separated in subspaces with... 

Here, as elsewhere, the subscripts i and a stand for electrons and nuclei respectively. The factor (S Si)/S(S + 1) is used to project the contribution from each open shell electron i onto the total spin angular momentum S. We remind ourselves that the effective Hamiltonian is constructed to operate within an electronic state with a given multiplicity (2S +1). 

Only the unpaired electrons contribute to the sums over i and j because only these electrons contribute to the total spin. The operator that represents this contribution in the effective Hamiltonian is designed to act only within the manifold of the state of interest,, S). In other words, it should not make any explicit reference to the quantum numbers S and S which must therefore be suppressed. With this in mind, the pair of 3-j symbols in equation (7.117) which both include S and S, can be re-expressed by making use of the relationship... 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

We begin with some general considerations of perhaps lesser-known, but important, features of exact electronic wavefunctions. Our motive is to establish a theoretical framework together with a reasonably consistent notation in order to carry through the spin-coupled VB and other expansions of the total wavefunction. We consider an atomic or molecular system consisting of N electrons and A nuclei. We assume the Born-Oppenheimer separation and write the Hamiltonian operator for the motion of the electrons in the form ... 

The theory of symmetry-preserving Kramers pair creation operators is reviewed and formulas for applying these operators to configuration interaction calculations are derived. A new and more general type of symmetry-preserving pair creation operator is proposed and shown to commute with the total spin operator and with all of the symmetry operations which leave the core Hamiltonian of a many-electron system invariant. The theory is extended to cases where orthonormality of orbitals of different configurations cannot be assumed. 

First let us discuss the case in which singlet and triplet states are well sep U ated in energy and axe represented by eigenstates of the Hamiltonian and the total spin operators 5 , Sz- It is verified iimnediately (noting that the functions (26) and (29) are normalized to unity on integrating over both variables) that... 

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